HardMathematicsHigh School (11–12)Coordinate Geometry › Circles and Conic SectionsNumerical

Two circular lakes have edges x² + y² - 4x - 6y - 12 = 0 and x² + y² - 16x - 6y + 48 = 0, with…

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Two circular lakes have edges $x^2 + y^2 - 4x - 6y - 12 = 0$ and $x^2 + y^2 - 16x - 6y + 48 = 0$ , with lengths in kilometres. They overlap. A bridge is built along the line through the two meeting points of their edges. Find the equation of the line in the form $x = k$ , the distance between the two centres, and the length of the bridge inside the lakes.

Need a reminder? The key ideaShow

Compare the distance $d$ between the centres with $r_1 + r_2$ and $|r_1 - r_2|$ . Then $4$ , $3$ , $2$ , $1$ or $0$ common tangents. A direct tangent has length $\sqrt{d^2 - (r_1 - r_2)^2}$ and a transverse tangent has length $\sqrt{d^2 - (r_1 + r_2)^2}$ . For two circles, $S_1 - S_2 = 0$ is the common chord or the common tangent.

The worked solution

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